23 - TrigSub

# 23 - TrigSub - Do 1 What is the appropriate trig change of...

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Trigonometric Substitution Suppose we have an integral of the form a 2 ! x 2 dx " . If u = a 2 ! x 2 then du = ! 2 x dx but we have no “x” in the integral. We cannot use u substitution. Instead we will substitute x = a sin ! with du = a cos ! d ! . a 2 ! x 2 dx " =

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Therefore: 1. If we have a 2 ! x 2 ( ) n 2 , n " 0, let x = a sin # 2. If we have a 2 + x 2 ( ) n 2 , n ! 0, let x = a tan " , dx = a sec 2 " d " Since 1 + tan 2 ! = sec 2 ! 3. If we have x 2 ! a 2 ( ) n 2 , n " 0, let x = a sec # , dx = a sec # tan # d # Since tan 2 ! = sec 2 ! " 1 Ex. Evaluate x 3 16 ! x 2 dx "
Ex. What integral is obtained from x 4 2 x 2 + 4 ( ) 3 2 ! dx ?

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Ex. What is the appropriate trig change of variable to simplify x 4 3 ! 4 x 2 dx " ? What is the new integral? Ex. What is the appropriate trig change of variable to simplify 2 x 2 ! 4 dx

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Unformatted text preview: Do: 1. What is the appropriate trig change of variable to simplify x 2 2 ! 5 x 2 ( ) 3 2 dx " 2. What new integral is obtained? Ex. Suppose I used x = 3sec ! to make the substitution and I ended with the following integral: + sin 2 ( ) + tan + C Convert the expression back to an expression in x. Ex. Suppose I used x = 5 tan to make the substitution and I ended with the following integral: cos !" 3sin ( ) + + C Convert the expression back to an expression in x....
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